EXTINCTION BY BAD LUCK Science Journaling Club, Volume 1 Issue 1, Fall 2024 A population that grows on average can still die out. master seed : 21091847 numpy version : 2.4.2 python : 3.12.3 replicates per cell : 20000 replicates, convergence run: 200000 mean offspring m : 1.1500 (held fixed across all variance levels) horizon, Model A : 100 generations escape cap, Model A : 20000 individuals ================================================================================================ SECTION 1. ANALYTIC EXTINCTION PROBABILITIES, AND THE CHECKS ON THEM ================================================================================================ For each offspring law: q is the smallest non-negative root of f(s) = s, found by bisection; q_G = f_G(0) is the exact probability that ONE lineage is extinct by generation 100. Closed forms are solved independently where they exist. offspring law mean var q (bisect) q_G=f_G(0) resid f(q)-q cap bias ------------------------------------------------------------------------------------------------ Poisson 1.1500 1.1500 0.75099795 0.75099787 0.00e+00 0.0e+00 Neg. binomial r=4 1.1500 1.4806 0.79705349 0.79705341 0.00e+00 0.0e+00 Geometric (NB r=1) 1.1500 2.4725 0.86956522 0.86956512 1.11e-16 0.0e+00 Neg. binomial r=0.25 1.1500 6.4400 0.94634805 0.94634798 0.00e+00 0.0e+00 Lottery K=30 1.1500 33.1775 0.99015744 0.99015744 1.11e-16 1.2e-86 ------------------------------------------------------------------------------------------------ How fast does extinction-within-G-generations approach eventual extinction? f_G(0) for one starting individual, by horizon length: offspring law G=5 G=10 G=20 G=50 G=100 q - f_100(0) ------------------------------------------------------------------------------------------------ Poisson 0.6209594 0.7025643 0.7413419 0.7508837 0.7509979 7.49e-08 Neg. binomial r=4 0.6749888 0.7520309 0.7879533 0.7969370 0.7970534 8.79e-08 Geometric (NB r=1) 0.7702288 0.8338531 0.8622463 0.8694605 0.8695651 9.66e-08 Neg. binomial r=0.25 0.8912739 0.9275510 0.9424934 0.9462870 0.9463480 6.81e-08 Lottery K=30 0.9837795 0.9879163 0.9897234 0.9901524 0.9901574 3.22e-09 ------------------------------------------------------------------------------------------------ Independent closed-form checks on the same roots: check club value closed form difference ------------------------------------------------------------------------------------------------ geometric: q = 1/m 0.8695652174 0.8695652174 -4.44e-16 Poisson: q = -W(-m e^-m)/m 0.7509979468 0.7509979468 -1.22e-15 geometric q_G vs 1/m limit 0.8695651208 0.8695652174 -9.66e-08 ------------------------------------------------------------------------------------------------ The last line is negative by construction: q_G is extinction within 100 generations and must sit below the eventual value q. At G = 100 the gap has closed to less than a part in a million for every law we use, so the horizon is not what sets our answer. At G = 20 it would be, which is why the table above is printed rather than assumed. ================================================================================================ SECTION 2. MODEL A, BRANCHING PROCESS: SIMULATED VS ANALYTIC, EVERY CELL ================================================================================================ P_sim is the fraction of 20000 replicates extinct within 100 generations. P_ana = (f_G(0))^N0 is exact. z = (P_sim - P_ana) / SE, SE from the binomial. CI is the 95 percent Wilson interval on P_sim. --- Poisson (offspring variance 1.1500, q = 0.750998, q_G = 0.750998) --- N0 P_sim 95% CI P_ana diff z mean T_ext n_ext ------------------------------------------------------------------------------------------------ 2 0.56935 [ 0.56247, 0.57620] 0.56400 0.00535 1.53 5.24 11387 3 0.42555 [ 0.41871, 0.43242] 0.42356 0.00199 0.57 6.77 8511 4 0.31570 [ 0.30929, 0.32218] 0.31809 -0.00239 -0.73 7.76 6314 5 0.23730 [ 0.23145, 0.24325] 0.23889 -0.00159 -0.53 8.78 4746 6 0.17880 [ 0.17355, 0.18417] 0.17940 -0.00060 -0.22 9.68 3576 7 0.13715 [ 0.13245, 0.14199] 0.13473 0.00242 1.00 10.51 2743 8 0.10380 [ 0.09965, 0.10810] 0.10118 0.00262 1.23 10.95 2076 9 0.07400 [ 0.07045, 0.07771] 0.07599 -0.00199 -1.06 11.78 1480 10 0.05800 [ 0.05484, 0.06133] 0.05707 0.00093 0.57 12.31 1160 11 0.04205 [ 0.03936, 0.04492] 0.04286 -0.00081 -0.56 12.25 841 12 0.03210 [ 0.02975, 0.03463] 0.03219 -0.00009 -0.07 13.15 642 13 0.02410 [ 0.02206, 0.02632] 0.02417 -0.00007 -0.07 13.29 482 14 0.01795 [ 0.01620, 0.01988] 0.01815 -0.00020 -0.21 14.15 359 15 0.01345 [ 0.01194, 0.01514] 0.01363 -0.00018 -0.22 14.43 269 16 0.00990 [ 0.00862, 0.01137] 0.01024 -0.00034 -0.47 15.28 198 17 0.00775 [ 0.00663, 0.00906] 0.00769 0.00006 0.10 15.59 155 18 0.00525 [ 0.00434, 0.00635] 0.00577 -0.00052 -0.98 15.69 105 19 0.00440 [ 0.00357, 0.00542] 0.00434 0.00006 0.14 16.84 88 20 0.00335 [ 0.00264, 0.00425] 0.00326 0.00009 0.23 14.73 67 22 0.00180 [ 0.00130, 0.00249] 0.00184 -0.00004 -0.12 16.39 36 24 0.00065 [ 0.00038, 0.00111] 0.00104 -0.00039 -1.70 19.15 13 26 0.00035 [ 0.00017, 0.00072] 0.00058 -0.00023 -1.37 14.00 7 28 0.00045 [ 0.00024, 0.00086] 0.00033 0.00012 0.94 19.67 9 30 0.00025 [ 0.00011, 0.00059] 0.00019 0.00006 0.67 23.00 5 33 0.00010 [ 0.00003, 0.00036] 0.00008 0.00002 0.34 16.50 2 36 0.00005 [ 0.00001, 0.00028] 0.00003 0.00002 0.41 12.00 1 40 0.00005 [ 0.00001, 0.00028] 0.00001 0.00004 1.71 17.00 1 45 0.00000 [ 0.00000, 0.00019] 0.00000 -0.00000 -0.23 - 0 50 0.00000 [ 0.00000, 0.00019] 0.00000 -0.00000 -0.11 - 0 56 0.00000 [ 0.00000, 0.00019] 0.00000 -0.00000 -0.05 - 0 63 0.00000 [ 0.00000, 0.00019] 0.00000 -0.00000 -0.02 - 0 70 0.00000 [ 0.00000, 0.00019] 0.00000 -0.00000 -0.01 - 0 80 0.00000 [ 0.00000, 0.00019] 0.00000 -0.00000 -0.00 - 0 90 0.00000 [ 0.00000, 0.00019] 0.00000 -0.00000 -0.00 - 0 100 0.00000 [ 0.00000, 0.00019] 0.00000 -0.00000 -0.00 - 0 120 0.00000 [ 0.00000, 0.00019] 0.00000 -0.00000 -0.00 - 0 150 0.00000 [ 0.00000, 0.00019] 0.00000 -0.00000 -0.00 - 0 200 0.00000 [ 0.00000, 0.00019] 0.00000 -0.00000 -0.00 - 0 250 0.00000 [ 0.00000, 0.00019] 0.00000 -0.00000 -0.00 - 0 300 0.00000 [ 0.00000, 0.00019] 0.00000 -0.00000 -0.00 - 0 400 0.00000 [ 0.00000, 0.00019] 0.00000 -0.00000 -0.00 - 0 500 0.00000 [ 0.00000, 0.00019] 0.00000 -0.00000 -0.00 - 0 ------------------------------------------------------------------------------------------------ --- Neg. binomial r=4 (offspring variance 1.4806, q = 0.797053, q_G = 0.797053) --- N0 P_sim 95% CI P_ana diff z mean T_ext n_ext ------------------------------------------------------------------------------------------------ 2 0.63390 [ 0.62720, 0.64055] 0.63529 -0.00139 -0.41 4.90 12678 3 0.50420 [ 0.49727, 0.51113] 0.50636 -0.00216 -0.61 6.16 10084 4 0.40170 [ 0.39493, 0.40851] 0.40360 -0.00190 -0.55 7.34 8034 5 0.32295 [ 0.31650, 0.32946] 0.32169 0.00126 0.38 8.36 6459 6 0.25560 [ 0.24960, 0.26169] 0.25640 -0.00080 -0.26 9.14 5112 7 0.20255 [ 0.19704, 0.20818] 0.20437 -0.00182 -0.64 9.57 4051 8 0.15935 [ 0.15434, 0.16449] 0.16289 -0.00354 -1.36 10.53 3187 9 0.12830 [ 0.12374, 0.13301] 0.12983 -0.00153 -0.65 11.09 2566 10 0.10480 [ 0.10063, 0.10912] 0.10348 0.00132 0.61 11.66 2096 11 0.08120 [ 0.07749, 0.08507] 0.08248 -0.00128 -0.66 12.14 1624 12 0.06220 [ 0.05894, 0.06563] 0.06574 -0.00354 -2.02 12.61 1244 13 0.05065 [ 0.04770, 0.05378] 0.05240 -0.00175 -1.11 12.61 1013 14 0.04115 [ 0.03848, 0.04399] 0.04177 -0.00062 -0.44 13.40 823 15 0.03360 [ 0.03119, 0.03619] 0.03329 0.00031 0.24 13.53 672 16 0.02705 [ 0.02489, 0.02939] 0.02653 0.00052 0.45 14.35 541 17 0.02095 [ 0.01906, 0.02303] 0.02115 -0.00020 -0.20 13.60 419 18 0.01640 [ 0.01473, 0.01826] 0.01686 -0.00046 -0.50 15.05 328 19 0.01345 [ 0.01194, 0.01514] 0.01344 0.00001 0.02 14.79 269 20 0.01160 [ 0.01021, 0.01318] 0.01071 0.00089 1.22 15.81 232 22 0.00780 [ 0.00667, 0.00912] 0.00680 0.00100 1.71 16.65 156 24 0.00460 [ 0.00375, 0.00564] 0.00432 0.00028 0.60 14.99 92 26 0.00325 [ 0.00255, 0.00414] 0.00275 0.00050 1.36 17.49 65 28 0.00200 [ 0.00147, 0.00272] 0.00174 0.00026 0.87 17.10 40 30 0.00120 [ 0.00081, 0.00179] 0.00111 0.00009 0.39 16.83 24 33 0.00040 [ 0.00020, 0.00079] 0.00056 -0.00016 -0.96 14.12 8 36 0.00035 [ 0.00017, 0.00072] 0.00028 0.00007 0.55 18.86 7 40 0.00010 [ 0.00003, 0.00036] 0.00011 -0.00001 -0.19 17.00 2 45 0.00005 [ 0.00001, 0.00028] 0.00004 0.00001 0.31 22.00 1 50 0.00005 [ 0.00001, 0.00028] 0.00001 0.00004 1.57 8.00 1 56 0.00000 [ 0.00000, 0.00019] 0.00000 -0.00000 -0.25 - 0 63 0.00000 [ 0.00000, 0.00019] 0.00000 -0.00000 -0.11 - 0 70 0.00000 [ 0.00000, 0.00019] 0.00000 -0.00000 -0.05 - 0 80 0.00000 [ 0.00000, 0.00019] 0.00000 -0.00000 -0.02 - 0 90 0.00000 [ 0.00000, 0.00019] 0.00000 -0.00000 -0.01 - 0 100 0.00000 [ 0.00000, 0.00019] 0.00000 -0.00000 -0.00 - 0 120 0.00000 [ 0.00000, 0.00019] 0.00000 -0.00000 -0.00 - 0 150 0.00000 [ 0.00000, 0.00019] 0.00000 -0.00000 -0.00 - 0 200 0.00000 [ 0.00000, 0.00019] 0.00000 -0.00000 -0.00 - 0 250 0.00000 [ 0.00000, 0.00019] 0.00000 -0.00000 -0.00 - 0 300 0.00000 [ 0.00000, 0.00019] 0.00000 -0.00000 -0.00 - 0 400 0.00000 [ 0.00000, 0.00019] 0.00000 -0.00000 -0.00 - 0 500 0.00000 [ 0.00000, 0.00019] 0.00000 -0.00000 -0.00 - 0 ------------------------------------------------------------------------------------------------ --- Geometric (NB r=1) (offspring variance 2.4725, q = 0.869565, q_G = 0.869565) --- N0 P_sim 95% CI P_ana diff z mean T_ext n_ext ------------------------------------------------------------------------------------------------ 2 0.75695 [ 0.75096, 0.76284] 0.75614 0.00081 0.27 4.16 15139 3 0.65660 [ 0.64999, 0.66315] 0.65752 -0.00092 -0.27 5.21 13132 4 0.56915 [ 0.56227, 0.57600] 0.57175 -0.00260 -0.74 6.21 11383 5 0.49460 [ 0.48767, 0.50153] 0.49718 -0.00258 -0.73 6.95 9892 6 0.43455 [ 0.42769, 0.44143] 0.43233 0.00222 0.63 7.70 8691 7 0.37790 [ 0.37120, 0.38464] 0.37594 0.00196 0.57 8.28 7558 8 0.31980 [ 0.31337, 0.32630] 0.32690 -0.00710 -2.14 8.86 6396 9 0.28905 [ 0.28281, 0.29537] 0.28426 0.00479 1.50 9.36 5781 10 0.24705 [ 0.24112, 0.25308] 0.24718 -0.00013 -0.04 9.98 4941 11 0.21580 [ 0.21015, 0.22156] 0.21494 0.00086 0.30 10.44 4316 12 0.18580 [ 0.18047, 0.19125] 0.18691 -0.00111 -0.40 10.90 3716 13 0.16215 [ 0.15711, 0.16732] 0.16253 -0.00038 -0.14 11.28 3243 14 0.13985 [ 0.13511, 0.14473] 0.14133 -0.00148 -0.60 11.62 2797 15 0.12560 [ 0.12108, 0.13026] 0.12289 0.00271 1.17 11.77 2512 16 0.10885 [ 0.10461, 0.11324] 0.10686 0.00199 0.91 12.36 2177 17 0.09345 [ 0.08949, 0.09756] 0.09293 0.00052 0.26 12.84 1869 18 0.08215 [ 0.07842, 0.08604] 0.08080 0.00135 0.70 12.92 1643 19 0.07145 [ 0.06796, 0.07510] 0.07027 0.00118 0.66 13.08 1429 20 0.05915 [ 0.05596, 0.06250] 0.06110 -0.00195 -1.15 13.40 1183 22 0.04420 [ 0.04144, 0.04714] 0.04620 -0.00200 -1.35 13.88 884 24 0.03525 [ 0.03278, 0.03790] 0.03493 0.00032 0.24 14.35 705 26 0.02505 [ 0.02297, 0.02731] 0.02642 -0.00137 -1.20 14.97 501 28 0.01890 [ 0.01710, 0.02088] 0.01997 -0.00107 -1.09 15.63 378 30 0.01560 [ 0.01397, 0.01741] 0.01510 0.00050 0.58 15.57 312 33 0.00855 [ 0.00737, 0.00992] 0.00993 -0.00138 -1.97 16.87 171 36 0.00750 [ 0.00640, 0.00879] 0.00653 0.00097 1.70 17.74 150 40 0.00400 [ 0.00322, 0.00498] 0.00373 0.00027 0.62 16.85 80 45 0.00185 [ 0.00134, 0.00255] 0.00186 -0.00001 -0.02 21.05 37 50 0.00105 [ 0.00069, 0.00160] 0.00092 0.00013 0.59 18.19 21 56 0.00050 [ 0.00027, 0.00092] 0.00040 0.00010 0.72 20.00 10 63 0.00005 [ 0.00001, 0.00028] 0.00015 -0.00010 -1.15 17.00 1 70 0.00015 [ 0.00005, 0.00044] 0.00006 0.00009 1.76 23.67 3 80 0.00000 [ 0.00000, 0.00019] 0.00001 -0.00001 -0.53 - 0 90 0.00000 [ 0.00000, 0.00019] 0.00000 -0.00000 -0.26 - 0 100 0.00000 [ 0.00000, 0.00019] 0.00000 -0.00000 -0.13 - 0 120 0.00000 [ 0.00000, 0.00019] 0.00000 -0.00000 -0.03 - 0 150 0.00000 [ 0.00000, 0.00019] 0.00000 -0.00000 -0.00 - 0 200 0.00000 [ 0.00000, 0.00019] 0.00000 -0.00000 -0.00 - 0 250 0.00000 [ 0.00000, 0.00019] 0.00000 -0.00000 -0.00 - 0 300 0.00000 [ 0.00000, 0.00019] 0.00000 -0.00000 -0.00 - 0 400 0.00000 [ 0.00000, 0.00019] 0.00000 -0.00000 -0.00 - 0 500 0.00000 [ 0.00000, 0.00019] 0.00000 -0.00000 -0.00 - 0 ------------------------------------------------------------------------------------------------ --- Neg. binomial r=0.25 (offspring variance 6.4400, q = 0.946348, q_G = 0.946348) --- N0 P_sim 95% CI P_ana diff z mean T_ext n_ext ------------------------------------------------------------------------------------------------ 2 0.89800 [ 0.89373, 0.90212] 0.89557 0.00243 1.12 2.85 17960 3 0.85010 [ 0.84509, 0.85498] 0.84753 0.00257 1.01 3.55 17002 4 0.80430 [ 0.79874, 0.80974] 0.80205 0.00225 0.80 4.22 16086 5 0.76410 [ 0.75817, 0.76993] 0.75902 0.00508 1.68 4.78 15282 6 0.71860 [ 0.71233, 0.72479] 0.71830 0.00030 0.09 5.25 14372 7 0.67965 [ 0.67315, 0.68608] 0.67976 -0.00011 -0.03 5.79 13593 8 0.63685 [ 0.63016, 0.64349] 0.64329 -0.00644 -1.90 6.17 12737 9 0.60535 [ 0.59856, 0.61210] 0.60878 -0.00343 -0.99 6.51 12107 10 0.57225 [ 0.56538, 0.57909] 0.57611 -0.00386 -1.11 6.79 11445 11 0.54890 [ 0.54199, 0.55579] 0.54520 0.00370 1.05 7.37 10978 12 0.51920 [ 0.51227, 0.52612] 0.51595 0.00325 0.92 7.60 10384 13 0.49155 [ 0.48462, 0.49848] 0.48827 0.00328 0.93 7.96 9831 14 0.46280 [ 0.45590, 0.46972] 0.46207 0.00073 0.21 8.30 9256 15 0.43805 [ 0.43119, 0.44494] 0.43728 0.00077 0.22 8.59 8761 16 0.41395 [ 0.40714, 0.42079] 0.41382 0.00013 0.04 8.77 8279 17 0.39310 [ 0.38635, 0.39989] 0.39162 0.00148 0.43 9.01 7862 18 0.37030 [ 0.36363, 0.37702] 0.37061 -0.00031 -0.09 9.39 7406 19 0.35110 [ 0.34451, 0.35774] 0.35072 0.00038 0.11 9.42 7022 20 0.32730 [ 0.32083, 0.33384] 0.33191 -0.00461 -1.38 9.88 6546 22 0.29685 [ 0.29056, 0.30322] 0.29725 -0.00040 -0.12 10.22 5937 24 0.26635 [ 0.26027, 0.27252] 0.26621 0.00014 0.05 10.83 5327 26 0.24105 [ 0.23517, 0.24703] 0.23841 0.00264 0.88 10.88 4821 28 0.21640 [ 0.21075, 0.22216] 0.21351 0.00289 1.00 11.27 4328 30 0.19005 [ 0.18467, 0.19555] 0.19122 -0.00117 -0.42 11.78 3801 33 0.16435 [ 0.15928, 0.16955] 0.16206 0.00229 0.88 12.10 3287 36 0.13675 [ 0.13206, 0.14158] 0.13735 -0.00060 -0.25 12.42 2735 40 0.10880 [ 0.10456, 0.11319] 0.11016 -0.00136 -0.62 13.38 2176 45 0.08430 [ 0.08053, 0.08823] 0.08362 0.00068 0.35 13.86 1686 50 0.06275 [ 0.05947, 0.06620] 0.06347 -0.00072 -0.42 14.53 1255 56 0.04380 [ 0.04105, 0.04672] 0.04559 -0.00179 -1.21 15.11 876 63 0.03210 [ 0.02975, 0.03463] 0.03099 0.00111 0.91 16.59 642 70 0.02195 [ 0.02001, 0.02407] 0.02106 0.00089 0.87 17.36 439 80 0.01195 [ 0.01054, 0.01355] 0.01214 -0.00019 -0.24 17.03 239 90 0.00700 [ 0.00594, 0.00825] 0.00699 0.00001 0.01 18.40 140 100 0.00395 [ 0.00317, 0.00492] 0.00403 -0.00008 -0.17 18.32 79 120 0.00090 [ 0.00057, 0.00142] 0.00134 -0.00044 -1.69 21.72 18 150 0.00025 [ 0.00011, 0.00059] 0.00026 -0.00001 -0.05 14.80 5 200 0.00000 [ 0.00000, 0.00019] 0.00002 -0.00002 -0.57 - 0 250 0.00000 [ 0.00000, 0.00019] 0.00000 -0.00000 -0.14 - 0 300 0.00000 [ 0.00000, 0.00019] 0.00000 -0.00000 -0.04 - 0 400 0.00000 [ 0.00000, 0.00019] 0.00000 -0.00000 -0.00 - 0 500 0.00000 [ 0.00000, 0.00019] 0.00000 -0.00000 -0.00 - 0 ------------------------------------------------------------------------------------------------ --- Lottery K=30 (offspring variance 33.1775, q = 0.990157, q_G = 0.990157) --- N0 P_sim 95% CI P_ana diff z mean T_ext n_ext ------------------------------------------------------------------------------------------------ 2 0.98270 [ 0.98080, 0.98442] 0.98041 0.00229 2.34 1.20 19654 3 0.97150 [ 0.96910, 0.97372] 0.97076 0.00074 0.62 1.29 19430 4 0.96200 [ 0.95926, 0.96456] 0.96121 0.00079 0.58 1.40 19240 5 0.94895 [ 0.94581, 0.95192] 0.95175 -0.00280 -1.85 1.49 18979 6 0.94110 [ 0.93775, 0.94428] 0.94238 -0.00128 -0.78 1.58 18822 7 0.93240 [ 0.92884, 0.93580] 0.93310 -0.00070 -0.40 1.67 18648 8 0.92105 [ 0.91723, 0.92471] 0.92392 -0.00287 -1.53 1.81 18421 9 0.91315 [ 0.90917, 0.91697] 0.91483 -0.00168 -0.85 1.86 18263 10 0.90805 [ 0.90397, 0.91198] 0.90582 0.00223 1.08 1.94 18161 11 0.89230 [ 0.88793, 0.89652] 0.89691 -0.00461 -2.14 2.07 17846 12 0.88750 [ 0.88305, 0.89180] 0.88808 -0.00058 -0.26 2.15 17750 13 0.87715 [ 0.87253, 0.88163] 0.87934 -0.00219 -0.95 2.22 17543 14 0.87020 [ 0.86547, 0.87479] 0.87068 -0.00048 -0.20 2.34 17404 15 0.86180 [ 0.85695, 0.86651] 0.86211 -0.00031 -0.13 2.42 17236 16 0.85455 [ 0.84960, 0.85937] 0.85363 0.00092 0.37 2.43 17091 17 0.84995 [ 0.84493, 0.85483] 0.84522 0.00473 1.85 2.51 16999 18 0.83360 [ 0.82837, 0.83870] 0.83691 -0.00331 -1.27 2.62 16672 19 0.82870 [ 0.82342, 0.83386] 0.82867 0.00003 0.01 2.72 16574 20 0.82160 [ 0.81623, 0.82684] 0.82051 0.00109 0.40 2.78 16432 22 0.80360 [ 0.79804, 0.80905] 0.80444 -0.00084 -0.30 2.96 16072 24 0.78740 [ 0.78167, 0.79301] 0.78868 -0.00128 -0.44 3.14 15748 26 0.78055 [ 0.77476, 0.78623] 0.77323 0.00732 2.47 3.21 15611 28 0.76010 [ 0.75413, 0.76597] 0.75809 0.00201 0.66 3.39 15202 30 0.73695 [ 0.73080, 0.74301] 0.74324 -0.00629 -2.04 3.51 14739 33 0.71625 [ 0.70996, 0.72246] 0.72151 -0.00526 -1.66 3.76 14325 36 0.70795 [ 0.70161, 0.71421] 0.70041 0.00754 2.33 3.95 14159 40 0.67120 [ 0.66466, 0.67768] 0.67324 -0.00204 -0.62 4.20 13424 45 0.63755 [ 0.63086, 0.64419] 0.64075 -0.00320 -0.94 4.49 12751 50 0.60865 [ 0.60187, 0.61539] 0.60984 -0.00119 -0.34 4.76 12173 56 0.57500 [ 0.56814, 0.58184] 0.57470 0.00030 0.09 5.21 11500 63 0.53470 [ 0.52778, 0.54161] 0.53625 -0.00155 -0.44 5.53 10694 70 0.50295 [ 0.49602, 0.50988] 0.50038 0.00257 0.73 5.79 10059 80 0.45300 [ 0.44611, 0.45991] 0.45325 -0.00025 -0.07 6.18 9060 90 0.41355 [ 0.40674, 0.42039] 0.41057 0.00298 0.86 6.62 8271 100 0.37070 [ 0.36403, 0.37742] 0.37190 -0.00120 -0.35 7.24 7414 120 0.30735 [ 0.30099, 0.31378] 0.30515 0.00220 0.68 7.76 6147 150 0.22720 [ 0.22145, 0.23306] 0.22680 0.00040 0.14 8.78 4544 200 0.13560 [ 0.13093, 0.14041] 0.13831 -0.00271 -1.11 10.22 2712 250 0.08280 [ 0.07906, 0.08670] 0.08435 -0.00155 -0.79 11.32 1656 300 0.05280 [ 0.04979, 0.05599] 0.05144 0.00136 0.87 12.09 1056 400 0.01860 [ 0.01682, 0.02057] 0.01913 -0.00053 -0.55 13.39 372 500 0.00725 [ 0.00617, 0.00852] 0.00711 0.00014 0.23 15.37 145 ------------------------------------------------------------------------------------------------ ================================================================================================ SECTION 3. DID THE SIMULATOR PASS? ================================================================================================ Every simulated cell has an exact analytic partner, so the z-scores above are a sample from the standard normal if and only if the simulator is correct. Cells where fewer than 10 extinctions are expected are held back from this pool, because the normal approximation to a binomial is worthless there. They are checked separately with an exact Poisson tail probability. cells compared : 154 mean z : -0.0482 (expected 0, SE of the mean 0.0806) sd of z : 0.9429 (expected 1) max |z| : 2.471 cells with |z| > 1.96 : 8 of 154 (expected about 7.7) cells with |z| > 3 : 0 of 154 (expected about 0.4) rare cells held back : 56 (fewer than 10 extinctions expected) smallest Poisson tail p : 0.2104 (Geometric (NB r=1), N0 = 70: 3 seen, 1.13 expected) rare cells with p < 0.001 : 0 of 56 Verdict: PASS, the measured curve is the analytic curve within Monte Carlo error. ================================================================================================ SECTION 4. THE THRESHOLD: HOW BIG IS BIG ENOUGH FOR 5 PERCENT? ================================================================================================ N* is the starting size at which extinction probability within 100 generations falls to 0.05. Analytic: N* = ln(0.05) / ln(q_G). Simulated: weighted least-squares fit of ln(P_sim) on N0 over the cells where P_sim is between 0.004 and 0.45, extrapolated to 0.05, with a delta-method standard error. N_grid is the smallest size on our grid whose entire 95 percent CI sits below 0.05. offspring law variance N* analytic N* sim SE diff N_grid ------------------------------------------------------------------------------------------------ Poisson 1.1500 10.46 10.45 0.03 -0.01 11 Neg. binomial r=4 1.4806 13.21 13.18 0.04 -0.02 14 Geometric (NB r=1) 2.4725 21.43 21.41 0.06 -0.03 22 Neg. binomial r=0.25 6.4400 54.32 54.36 0.18 0.03 56 Lottery K=30 33.1775 302.86 302.05 1.47 -0.81 400 ------------------------------------------------------------------------------------------------ Same threshold, but for EVENTUAL extinction rather than extinction within the horizon. This is where the choice of horizon shows up in the answer. offspring law variance N* (100 gen) N* (eventual) ratio ------------------------------------------------------------------------------------------------ Poisson 1.1500 10.46 10.46 1.000 Neg. binomial r=4 1.4806 13.21 13.21 1.000 Geometric (NB r=1) 2.4725 21.43 21.43 1.000 Neg. binomial r=0.25 6.4400 54.32 54.32 1.000 Lottery K=30 33.1775 302.86 302.86 1.000 ------------------------------------------------------------------------------------------------ Threshold against variance, as a scaling: N* rises close to linearly in the offspring variance once the variance is well above the mean. offspring law variance N* (100 gen) N*/variance ------------------------------------------------------------------------------------------------ Poisson 1.1500 10.46 9.097 Neg. binomial r=4 1.4806 13.21 8.920 Geometric (NB r=1) 2.4725 21.43 8.669 Neg. binomial r=0.25 6.4400 54.32 8.436 Lottery K=30 33.1775 302.86 9.129 ------------------------------------------------------------------------------------------------ log-log fit over the five levels: ln N* = 2.1781 + 1.0014 ln(variance), R^2 = 0.99940 A slope of 1 would mean the threshold is exactly proportional to variance. ================================================================================================ SECTION 5. TIME TO EXTINCTION, AMONG THE POPULATIONS THAT DIED ================================================================================================ Conditioned on dying inside the horizon. Generations, mean and its standard error, with quartiles and the 95th percentile of the same conditional distribution. Blank rows are cells where fewer than 30 replicates died. offspring law N0 n_extinct mean SE q25 median q75 p95 ------------------------------------------------------------------------------------------------ Poisson 2 11306 5.238 0.051 2.0 3.0 7.0 16.0 Poisson 5 4745 8.909 0.097 4.0 7.0 11.0 22.0 Poisson 10 1151 12.154 0.225 7.0 10.0 15.5 26.0 Poisson 20 54 15.093 1.195 10.0 13.5 18.0 25.7 Poisson 50 0 - - - - - - Poisson 100 0 - - - - - - Poisson 200 0 - - - - - - Poisson 500 0 - - - - - - ------------------------------------------------------------------------------------------------ Neg. binomial r=4 2 12872 4.903 0.046 2.0 3.0 6.0 15.0 Neg. binomial r=4 5 6549 8.353 0.083 4.0 6.0 11.0 21.0 Neg. binomial r=4 10 2094 11.368 0.167 6.0 9.0 14.0 27.0 Neg. binomial r=4 20 218 15.234 0.529 9.2 13.0 18.8 31.0 Neg. binomial r=4 50 0 - - - - - - Neg. binomial r=4 100 0 - - - - - - Neg. binomial r=4 200 0 - - - - - - Neg. binomial r=4 500 0 - - - - - - ------------------------------------------------------------------------------------------------ Geometric (NB r=1) 2 15207 4.169 0.038 1.0 2.0 5.0 14.0 Geometric (NB r=1) 5 9927 6.929 0.061 3.0 5.0 9.0 19.0 Geometric (NB r=1) 10 4970 9.771 0.101 5.0 8.0 12.0 23.0 Geometric (NB r=1) 20 1213 13.344 0.220 8.0 11.0 17.0 29.0 Geometric (NB r=1) 50 12 - - - - - - Geometric (NB r=1) 100 0 - - - - - - Geometric (NB r=1) 200 0 - - - - - - Geometric (NB r=1) 500 0 - - - - - - ------------------------------------------------------------------------------------------------ Neg. binomial r=0.25 2 17938 2.771 0.026 1.0 2.0 3.0 9.0 Neg. binomial r=0.25 5 15145 4.706 0.040 2.0 3.0 6.0 14.0 Neg. binomial r=0.25 10 11463 6.913 0.056 3.0 5.0 9.0 19.0 Neg. binomial r=0.25 20 6659 9.805 0.085 5.0 8.0 12.0 23.0 Neg. binomial r=0.25 50 1277 14.660 0.231 9.0 13.0 18.0 31.0 Neg. binomial r=0.25 100 74 18.622 0.983 12.0 16.0 24.0 34.3 Neg. binomial r=0.25 200 0 - - - - - - Neg. binomial r=0.25 500 0 - - - - - - ------------------------------------------------------------------------------------------------ Lottery K=30 2 19620 1.208 0.010 1.0 1.0 1.0 2.0 Lottery K=30 5 19048 1.490 0.015 1.0 1.0 1.0 4.0 Lottery K=30 10 18146 1.945 0.020 1.0 1.0 2.0 6.0 Lottery K=30 20 16344 2.769 0.029 1.0 1.0 3.0 10.0 Lottery K=30 50 12153 4.754 0.046 2.0 3.0 6.0 15.0 Lottery K=30 100 7465 7.206 0.073 3.0 5.0 9.0 20.0 Lottery K=30 200 2787 10.234 0.136 5.0 8.0 13.0 24.0 Lottery K=30 500 110 14.936 0.800 9.0 13.0 18.0 31.1 ------------------------------------------------------------------------------------------------ ================================================================================================ SECTION 6. CONVERGENCE OF THE MONTE CARLO ESTIMATE ================================================================================================ One cell run to 200000 replicates, estimate recorded as trials accumulate. halfwidth is the 95 percent normal-approximation halfwidth at that trial count. --- Geometric (NB r=1), N0 = 20, analytic P = 0.061100 --- trials estimate halfwidth error err/SE ------------------------------------------------------------------------------------------------ 100 0.050000 0.042716 -0.011100 -0.46 200 0.060000 0.032913 -0.001100 -0.06 500 0.058000 0.020488 -0.003100 -0.29 1000 0.055000 0.014130 -0.006100 -0.81 2000 0.053000 0.009819 -0.008100 -1.51 5000 0.060400 0.006603 -0.000700 -0.21 10000 0.064100 0.004801 +0.003000 1.25 20000 0.062000 0.003342 +0.000900 0.53 50000 0.061180 0.002101 +0.000080 0.07 100000 0.060600 0.001479 -0.000500 -0.66 200000 0.061390 0.001052 +0.000290 0.54 ------------------------------------------------------------------------------------------------ --- Poisson, N0 = 10, analytic P = 0.057067 --- trials estimate halfwidth error err/SE ------------------------------------------------------------------------------------------------ 100 0.030000 0.033434 -0.027067 -1.17 200 0.050000 0.030205 -0.007067 -0.43 500 0.048000 0.018737 -0.009067 -0.87 1000 0.049000 0.013379 -0.008067 -1.10 2000 0.056000 0.010077 -0.001067 -0.21 5000 0.055000 0.006319 -0.002067 -0.63 10000 0.054700 0.004457 -0.002367 -1.02 20000 0.055550 0.003174 -0.001517 -0.93 50000 0.055980 0.002015 -0.001087 -1.05 100000 0.056290 0.001429 -0.000777 -1.06 200000 0.056840 0.001015 -0.000227 -0.44 ------------------------------------------------------------------------------------------------ --- Lottery K=30, N0 = 120, analytic P = 0.305148 --- trials estimate halfwidth error err/SE ------------------------------------------------------------------------------------------------ 100 0.290000 0.088936 -0.015148 -0.33 200 0.270000 0.061529 -0.035148 -1.08 500 0.284000 0.039526 -0.021148 -1.03 1000 0.287000 0.028037 -0.018148 -1.25 2000 0.295500 0.019996 -0.009648 -0.94 5000 0.314200 0.012867 +0.009052 1.39 10000 0.311300 0.009075 +0.006152 1.34 20000 0.306150 0.006388 +0.001002 0.31 50000 0.304360 0.004033 -0.000788 -0.38 100000 0.303540 0.002850 -0.001608 -1.10 200000 0.304200 0.002016 -0.000948 -0.92 ------------------------------------------------------------------------------------------------ ================================================================================================ SECTION 7. MODEL B, CONTINUOUS-TIME BIRTH-DEATH WITH RISING TURNOVER ================================================================================================ Every level has the same Malthusian growth rate r = lambda - mu = 0.15 per unit time. Only the turnover lambda + mu changes, which is the demographic variance rate per individual. Horizon t = 100, 20000 replicates per cell. The escape cap is set per level so that the extinction probability still available to a capped replicate is below 1e-6, and never below twice the largest start size. P_ana = [mu(1 - e^{-rt}) / (lambda - mu e^{-rt})]^{N0}, exact. --- lambda = 0.65, mu = 0.50, turnover 1.15, eventual q = 0.769231, cap 160, cap bias 5.9e-19 --- N0 P_sim 95% CI P_ana diff z mean t_ext n_ext ------------------------------------------------------------------------------------------------ 2 0.59575 [ 0.58893, 0.60253] 0.59172 0.00403 1.16 4.658 11915 3 0.45640 [ 0.44951, 0.46331] 0.45517 0.00123 0.35 6.056 9128 5 0.26740 [ 0.26131, 0.27358] 0.26933 -0.00193 -0.61 8.232 5348 8 0.12115 [ 0.11670, 0.12575] 0.12259 -0.00144 -0.62 10.330 2423 12 0.04545 [ 0.04265, 0.04843] 0.04292 0.00253 1.76 12.387 909 20 0.00630 [ 0.00529, 0.00750] 0.00526 0.00104 2.03 14.587 126 30 0.00035 [ 0.00017, 0.00072] 0.00038 -0.00003 -0.23 28.973 7 45 0.00010 [ 0.00003, 0.00036] 0.00001 0.00009 4.79 14.702 2 60 0.00000 [ 0.00000, 0.00019] 0.00000 -0.00000 -0.05 - 0 80 0.00000 [ 0.00000, 0.00019] 0.00000 -0.00000 -0.00 - 0 ------------------------------------------------------------------------------------------------ threshold N* for 5 percent: analytic 11.42, simulated 11.58 (SE 0.08) --- lambda = 1.15, mu = 1.00, turnover 2.15, eventual q = 0.869565, cap 160, cap bias 1.9e-10 --- N0 P_sim 95% CI P_ana diff z mean t_ext n_ext ------------------------------------------------------------------------------------------------ 2 0.75220 [ 0.74617, 0.75813] 0.75614 -0.00394 -1.30 3.393 15044 3 0.65715 [ 0.65054, 0.66370] 0.65752 -0.00037 -0.11 4.466 13143 5 0.49920 [ 0.49227, 0.50613] 0.49718 0.00202 0.57 6.062 9984 8 0.32585 [ 0.31939, 0.33238] 0.32690 -0.00105 -0.32 7.797 6517 12 0.18490 [ 0.17958, 0.19034] 0.18691 -0.00201 -0.73 9.610 3698 20 0.06455 [ 0.06123, 0.06804] 0.06110 0.00345 2.04 12.185 1291 30 0.01460 [ 0.01303, 0.01636] 0.01510 -0.00050 -0.58 14.126 292 45 0.00165 [ 0.00118, 0.00232] 0.00186 -0.00021 -0.68 19.919 33 60 0.00025 [ 0.00011, 0.00059] 0.00023 0.00002 0.21 19.149 5 80 0.00000 [ 0.00000, 0.00019] 0.00001 -0.00001 -0.53 - 0 ------------------------------------------------------------------------------------------------ threshold N* for 5 percent: analytic 21.43, simulated 21.56 (SE 0.15) --- lambda = 2.15, mu = 2.00, turnover 4.15, eventual q = 0.930233, cap 192, cap bias 9.3e-07 --- N0 P_sim 95% CI P_ana diff z mean t_ext n_ext ------------------------------------------------------------------------------------------------ 2 0.86555 [ 0.86075, 0.87021] 0.86533 0.00022 0.09 2.240 17311 3 0.80340 [ 0.79783, 0.80885] 0.80496 -0.00156 -0.56 3.037 16068 5 0.69635 [ 0.68994, 0.70268] 0.69656 -0.00021 -0.06 4.168 13927 8 0.55570 [ 0.54880, 0.56258] 0.56070 -0.00500 -1.43 5.636 11114 12 0.41995 [ 0.41313, 0.42680] 0.41985 0.00010 0.03 7.047 8399 20 0.23730 [ 0.23145, 0.24325] 0.23541 0.00189 0.63 9.094 4746 30 0.11675 [ 0.11237, 0.12127] 0.11422 0.00253 1.12 11.118 2335 45 0.03925 [ 0.03665, 0.04203] 0.03860 0.00065 0.48 13.170 785 60 0.01285 [ 0.01138, 0.01451] 0.01305 -0.00020 -0.24 14.845 257 80 0.00330 [ 0.00259, 0.00420] 0.00307 0.00023 0.58 17.321 66 ------------------------------------------------------------------------------------------------ threshold N* for 5 percent: analytic 41.42, simulated 41.66 (SE 0.25) --- lambda = 3.15, mu = 3.00, turnover 6.15, eventual q = 0.952381, cap 284, cap bias 9.6e-07 --- N0 P_sim 95% CI P_ana diff z mean t_ext n_ext ------------------------------------------------------------------------------------------------ 2 0.90735 [ 0.90325, 0.91129] 0.90703 0.00032 0.16 1.792 18147 3 0.86870 [ 0.86395, 0.87331] 0.86384 0.00486 2.01 2.315 17374 5 0.78185 [ 0.77607, 0.78752] 0.78353 -0.00168 -0.58 3.330 15637 8 0.67750 [ 0.67099, 0.68394] 0.67684 0.00066 0.20 4.446 13550 12 0.55705 [ 0.55016, 0.56392] 0.55684 0.00021 0.06 5.707 11141 20 0.37415 [ 0.36747, 0.38088] 0.37689 -0.00274 -0.80 7.690 7483 30 0.22365 [ 0.21793, 0.22948] 0.23138 -0.00773 -2.59 9.125 4473 45 0.10760 [ 0.10338, 0.11197] 0.11130 -0.00370 -1.66 11.148 2152 60 0.05415 [ 0.05110, 0.05737] 0.05354 0.00061 0.39 12.396 1083 80 0.01825 [ 0.01649, 0.02020] 0.02018 -0.00193 -1.94 13.736 365 ------------------------------------------------------------------------------------------------ threshold N* for 5 percent: analytic 61.40, simulated 60.61 (SE 0.37) Model B validation, 34 cells with at least 10 expected extinctions: mean z -0.0340, sd 1.1025, max |z| 2.591, |z| above 1.96 in 4 cells. 6 rarer cells checked by exact Poisson tail. Smallest p = 0.0201 (lam 0.65, N0 = 45: 2 seen against 0.15 expected). Rare cells with p < 0.001: 0. Model B thresholds against turnover: lambda mu turnover eventual q N* analytic N* sim ------------------------------------------------------------------------------------------------ 0.65 0.50 1.15 0.769231 11.42 11.58 1.15 1.00 2.15 0.869565 21.43 21.56 2.15 2.00 4.15 0.930233 41.42 41.66 3.15 3.00 6.15 0.952381 61.40 60.61 ------------------------------------------------------------------------------------------------ ================================================================================================ SECTION 8. SENSITIVITY: WHAT IF THE MEAN IS DIFFERENT? ================================================================================================ Everything above fixes m = 1.15. Here is the analytic threshold N* for extinction within 100 generations, across a range of mean growth rates, for three of the offspring laws. Analytic only, no simulation needed. mean m Poisson N* Geometric N* Lottery K=30 N* ------------------------------------------------------------------------------------------------ 1.02 65.53 130.60 1900.80 1.05 30.25 60.94 876.89 1.10 15.46 31.43 447.86 1.15 10.46 21.43 302.86 1.25 6.45 13.43 186.61 1.50 3.43 7.39 98.80 2.00 1.88 4.32 53.86 ------------------------------------------------------------------------------------------------ The threshold explodes as the mean approaches replacement. That is the honest caveat on any single number we quote: N* depends on how far above 1 the mean is at least as strongly as it depends on the variance. ================================================================================================ SECTION 8B. AGAINST THE CLASSICAL APPROXIMATION (HALDANE 1927) ================================================================================================ Haldane's branching-process argument for a rare advantageous type gives the survival probability of a single lineage as roughly 2s / V, with s = m - 1 the excess growth and V the offspring variance. It is derived for small s. Our exact roots let us say how wrong it is at s = 0.15. offspring law variance 1 - q exact 2s/V approx ratio N* exact N* approx ------------------------------------------------------------------------------------------------ Poisson 1.1500 0.249002 0.260870 1.0477 10.46 11.48 Neg. binomial r=4 1.4806 0.202947 0.202617 0.9984 13.21 14.79 Geometric (NB r=1) 2.4725 0.130435 0.121335 0.9302 21.43 24.69 Neg. binomial r=0.25 6.4400 0.053652 0.046584 0.8683 54.32 64.31 Lottery K=30 33.1775 0.009843 0.009042 0.9187 302.86 331.30 ------------------------------------------------------------------------------------------------ The approximation sits within 0.2 to 13.2 percent of the exact survival probability across a variance range of nearly thirty, which is why the linear scaling of N* with variance that we measure was predictable from theory a century old. What the simulation adds is the size of the error in that theory, and the shape of the whole extinction curve rather than its rate alone. ================================================================================================ SECTION 9. NUMBERS THE ARTICLE QUOTES ================================================================================================ Mean offspring fixed at : 1.15 Poisson threshold N* (5%, 100 generations) : 10.5 Geometric threshold N* : 21.4 Lottery K=30 threshold N* : 302.9 Ratio, highest to lowest variance threshold : 28.95 Variance ratio over the same span : 28.85 Extinction probability, geometric, N0 = 2 : 0.7570 Extinction probability, geometric, N0 = 50 : 0.0010 Extinction probability, Poisson, N0 = 20 : 0.0034 Extinction probability, lottery, N0 = 100 : 0.3707 Largest |z| anywhere in Model A : 2.471 over 154 cells Largest |z| anywhere in Model B : 2.591 over 34 cells Runtime so far : 174.5 s ================================================================================================ SECTION 10. MACHINE-READABLE BLOCK (used to draw the figures) ================================================================================================ #BEGIN CURVES key,variance,N0,P_sim,lo,hi,P_ana pois,1.1500,2,0.569350,0.562475,0.576199,0.563998 pois,1.1500,3,0.425550,0.418713,0.432416,0.423561 pois,1.1500,4,0.315700,0.309294,0.322176,0.318094 pois,1.1500,5,0.237300,0.231455,0.243246,0.238888 pois,1.1500,6,0.178800,0.173551,0.184172,0.179404 pois,1.1500,7,0.137150,0.132452,0.141987,0.134732 pois,1.1500,8,0.103800,0.099649,0.108103,0.101183 pois,1.1500,9,0.074000,0.070453,0.077710,0.075989 pois,1.1500,10,0.058000,0.054845,0.061325,0.057067 pois,1.1500,11,0.042050,0.039355,0.044921,0.042857 pois,1.1500,12,0.032100,0.029746,0.034634,0.032186 pois,1.1500,13,0.024100,0.022064,0.026319,0.024171 pois,1.1500,14,0.017950,0.016200,0.019885,0.018153 pois,1.1500,15,0.013450,0.011944,0.015142,0.013633 pois,1.1500,16,0.009900,0.008619,0.011369,0.010238 pois,1.1500,17,0.007750,0.006626,0.009063,0.007689 pois,1.1500,18,0.005250,0.004339,0.006351,0.005774 pois,1.1500,19,0.004400,0.003573,0.005417,0.004336 pois,1.1500,20,0.003350,0.002639,0.004252,0.003257 pois,1.1500,22,0.001800,0.001301,0.002491,0.001837 pois,1.1500,24,0.000650,0.000380,0.001112,0.001036 pois,1.1500,26,0.000350,0.000170,0.000722,0.000584 pois,1.1500,28,0.000450,0.000237,0.000855,0.000330 pois,1.1500,30,0.000250,0.000107,0.000585,0.000186 pois,1.1500,33,0.000100,0.000027,0.000365,0.000079 pois,1.1500,36,0.000050,0.000009,0.000283,0.000033 pois,1.1500,40,0.000050,0.000009,0.000283,0.000011 pois,1.1500,45,0.000000,0.000000,0.000192,0.000003 pois,1.1500,50,0.000000,0.000000,0.000192,0.000001 pois,1.1500,56,0.000000,0.000000,0.000192,0.000000 pois,1.1500,63,0.000000,0.000000,0.000192,0.000000 pois,1.1500,70,0.000000,0.000000,0.000192,0.000000 pois,1.1500,80,0.000000,0.000000,0.000192,0.000000 pois,1.1500,90,0.000000,0.000000,0.000192,0.000000 pois,1.1500,100,0.000000,0.000000,0.000192,0.000000 pois,1.1500,120,0.000000,0.000000,0.000192,0.000000 pois,1.1500,150,0.000000,0.000000,0.000192,0.000000 pois,1.1500,200,0.000000,0.000000,0.000192,0.000000 pois,1.1500,250,0.000000,0.000000,0.000192,0.000000 pois,1.1500,300,0.000000,0.000000,0.000192,0.000000 pois,1.1500,400,0.000000,0.000000,0.000192,0.000000 pois,1.1500,500,0.000000,0.000000,0.000192,0.000000 nb4,1.4806,2,0.633900,0.627198,0.640550,0.635294 nb4,1.4806,3,0.504200,0.497271,0.511128,0.506363 nb4,1.4806,4,0.401700,0.394925,0.408513,0.403599 nb4,1.4806,5,0.322950,0.316504,0.329464,0.321690 nb4,1.4806,6,0.255600,0.249602,0.261692,0.256404 nb4,1.4806,7,0.202550,0.197037,0.208177,0.204368 nb4,1.4806,8,0.159350,0.154343,0.164488,0.162892 nb4,1.4806,9,0.128300,0.123736,0.133006,0.129834 nb4,1.4806,10,0.104800,0.100631,0.109121,0.103484 nb4,1.4806,11,0.081200,0.077494,0.085066,0.082482 nb4,1.4806,12,0.062200,0.058936,0.065632,0.065743 nb4,1.4806,13,0.050650,0.047696,0.053776,0.052401 nb4,1.4806,14,0.041150,0.038484,0.043992,0.041766 nb4,1.4806,15,0.033600,0.031191,0.036188,0.033290 nb4,1.4806,16,0.027050,0.024891,0.029391,0.026534 nb4,1.4806,17,0.020950,0.019055,0.023029,0.021149 nb4,1.4806,18,0.016400,0.014730,0.018255,0.016857 nb4,1.4806,19,0.013450,0.011944,0.015142,0.013436 nb4,1.4806,20,0.011600,0.010207,0.013181,0.010709 nb4,1.4806,22,0.007800,0.006672,0.009117,0.006803 nb4,1.4806,24,0.004600,0.003753,0.005638,0.004322 nb4,1.4806,26,0.003250,0.002551,0.004140,0.002746 nb4,1.4806,28,0.002000,0.001469,0.002722,0.001744 nb4,1.4806,30,0.001200,0.000807,0.001785,0.001108 nb4,1.4806,33,0.000400,0.000203,0.000789,0.000561 nb4,1.4806,36,0.000350,0.000170,0.000722,0.000284 nb4,1.4806,40,0.000100,0.000027,0.000365,0.000115 nb4,1.4806,45,0.000050,0.000009,0.000283,0.000037 nb4,1.4806,50,0.000050,0.000009,0.000283,0.000012 nb4,1.4806,56,0.000000,0.000000,0.000192,0.000003 nb4,1.4806,63,0.000000,0.000000,0.000192,0.000001 nb4,1.4806,70,0.000000,0.000000,0.000192,0.000000 nb4,1.4806,80,0.000000,0.000000,0.000192,0.000000 nb4,1.4806,90,0.000000,0.000000,0.000192,0.000000 nb4,1.4806,100,0.000000,0.000000,0.000192,0.000000 nb4,1.4806,120,0.000000,0.000000,0.000192,0.000000 nb4,1.4806,150,0.000000,0.000000,0.000192,0.000000 nb4,1.4806,200,0.000000,0.000000,0.000192,0.000000 nb4,1.4806,250,0.000000,0.000000,0.000192,0.000000 nb4,1.4806,300,0.000000,0.000000,0.000192,0.000000 nb4,1.4806,400,0.000000,0.000000,0.000192,0.000000 nb4,1.4806,500,0.000000,0.000000,0.000192,0.000000 geom,2.4725,2,0.756950,0.750957,0.762845,0.756143 geom,2.4725,3,0.656600,0.649990,0.663150,0.657516 geom,2.4725,4,0.569150,0.562274,0.575999,0.571753 geom,2.4725,5,0.494600,0.487673,0.501529,0.497176 geom,2.4725,6,0.434550,0.427693,0.441432,0.432327 geom,2.4725,7,0.377900,0.371204,0.384643,0.375937 geom,2.4725,8,0.319800,0.313371,0.326298,0.326901 geom,2.4725,9,0.289050,0.282808,0.295373,0.284262 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lot30,33.1775,200,0.135600,0.130925,0.140415,0.138309 lot30,33.1775,250,0.082800,0.079060,0.086700,0.084346 lot30,33.1775,300,0.052800,0.049786,0.055986,0.051437 lot30,33.1775,400,0.018600,0.016818,0.020567,0.019129 lot30,33.1775,500,0.007250,0.006165,0.008524,0.007114 #END CURVES #BEGIN THRESH key,variance,N_ana,N_sim,SE,N_ultimate pois,1.1500,10.4617,10.4509,0.0326,10.4617 nb4,1.4806,13.2067,13.1824,0.0383,13.2067 geom,2.4725,21.4345,21.4091,0.0604,21.4345 nb025,6.4400,54.3247,54.3585,0.1808,54.3248 lot30,33.1775,302.8647,302.0532,1.4717,302.8648 #END THRESH #BEGIN CONV key,n0,trials,estimate,halfwidth,analytic geom,20,100,0.050000,0.042716,0.061100 geom,20,200,0.060000,0.032913,0.061100 geom,20,500,0.058000,0.020488,0.061100 geom,20,1000,0.055000,0.014130,0.061100 geom,20,2000,0.053000,0.009819,0.061100 geom,20,5000,0.060400,0.006603,0.061100 geom,20,10000,0.064100,0.004801,0.061100 geom,20,20000,0.062000,0.003342,0.061100 geom,20,50000,0.061180,0.002101,0.061100 geom,20,100000,0.060600,0.001479,0.061100 geom,20,200000,0.061390,0.001052,0.061100 pois,10,100,0.030000,0.033434,0.057067 pois,10,200,0.050000,0.030205,0.057067 pois,10,500,0.048000,0.018737,0.057067 pois,10,1000,0.049000,0.013379,0.057067 pois,10,2000,0.056000,0.010077,0.057067 pois,10,5000,0.055000,0.006319,0.057067 pois,10,10000,0.054700,0.004457,0.057067 pois,10,20000,0.055550,0.003174,0.057067 pois,10,50000,0.055980,0.002015,0.057067 pois,10,100000,0.056290,0.001429,0.057067 pois,10,200000,0.056840,0.001015,0.057067 lot30,120,100,0.290000,0.088936,0.305148 lot30,120,200,0.270000,0.061529,0.305148 lot30,120,500,0.284000,0.039526,0.305148 lot30,120,1000,0.287000,0.028037,0.305148 lot30,120,2000,0.295500,0.019996,0.305148 lot30,120,5000,0.314200,0.012867,0.305148 lot30,120,10000,0.311300,0.009075,0.305148 lot30,120,20000,0.306150,0.006388,0.305148 lot30,120,50000,0.304360,0.004033,0.305148 lot30,120,100000,0.303540,0.002850,0.305148 lot30,120,200000,0.304200,0.002016,0.305148 #END CONV #BEGIN TIMES key,N0,n_extinct,mean,se,q25,median,q75,p95 pois,2,11306,5.2385,0.0507,2.0,3.0,7.0,16.0 pois,5,4745,8.9085,0.0973,4.0,7.0,11.0,22.0 pois,10,1151,12.1538,0.2247,7.0,10.0,15.5,26.0 pois,20,54,15.0926,1.1952,10.0,13.5,18.0,25.7 nb4,2,12872,4.9030,0.0464,2.0,3.0,6.0,15.0 nb4,5,6549,8.3527,0.0829,4.0,6.0,11.0,21.0 nb4,10,2094,11.3677,0.1672,6.0,9.0,14.0,27.0 nb4,20,218,15.2339,0.5289,9.2,13.0,18.8,31.0 geom,2,15207,4.1685,0.0383,1.0,2.0,5.0,14.0 geom,5,9927,6.9294,0.0606,3.0,5.0,9.0,19.0 geom,10,4970,9.7710,0.1009,5.0,8.0,12.0,23.0 geom,20,1213,13.3438,0.2196,8.0,11.0,17.0,29.0 nb025,2,17938,2.7710,0.0258,1.0,2.0,3.0,9.0 nb025,5,15145,4.7061,0.0396,2.0,3.0,6.0,14.0 nb025,10,11463,6.9130,0.0565,3.0,5.0,9.0,19.0 nb025,20,6659,9.8049,0.0849,5.0,8.0,12.0,23.0 nb025,50,1277,14.6601,0.2308,9.0,13.0,18.0,31.0 nb025,100,74,18.6216,0.9830,12.0,16.0,24.0,34.3 lot30,2,19620,1.2083,0.0100,1.0,1.0,1.0,2.0 lot30,5,19048,1.4901,0.0148,1.0,1.0,1.0,4.0 lot30,10,18146,1.9448,0.0204,1.0,1.0,2.0,6.0 lot30,20,16344,2.7690,0.0291,1.0,1.0,3.0,10.0 lot30,50,12153,4.7543,0.0462,2.0,3.0,6.0,15.0 lot30,100,7465,7.2058,0.0726,3.0,5.0,9.0,20.0 lot30,200,2787,10.2339,0.1356,5.0,8.0,13.0,24.0 lot30,500,110,14.9364,0.8005,9.0,13.0,18.0,31.1 #END TIMES #BEGIN BD lambda,mu,turnover,N0,P_sim,lo,hi,P_ana,mean_t_ext 0.65,0.50,1.15,2,0.595750,0.588931,0.602532,0.591716,4.6583 0.65,0.50,1.15,3,0.456400,0.449506,0.463311,0.455166,6.0557 0.65,0.50,1.15,5,0.267400,0.261311,0.273578,0.269329,8.2324 0.65,0.50,1.15,8,0.121150,0.116700,0.125745,0.122589,10.3304 0.65,0.50,1.15,12,0.045450,0.042650,0.048425,0.042922,12.3871 0.65,0.50,1.15,20,0.006300,0.005294,0.007495,0.005262,14.5869 0.65,0.50,1.15,30,0.000350,0.000170,0.000722,0.000382,28.9734 0.65,0.50,1.15,45,0.000100,0.000027,0.000365,0.000007,14.7015 0.65,0.50,1.15,60,0.000000,0.000000,0.000192,0.000000,nan 0.65,0.50,1.15,80,0.000000,0.000000,0.000192,0.000000,nan 1.15,1.00,2.15,2,0.752200,0.746169,0.758135,0.756144,3.3926 1.15,1.00,2.15,3,0.657150,0.650542,0.663698,0.657516,4.4662 1.15,1.00,2.15,5,0.499200,0.492271,0.506129,0.497177,6.0616 1.15,1.00,2.15,8,0.325850,0.319388,0.332379,0.326902,7.7971 1.15,1.00,2.15,12,0.184900,0.179580,0.190341,0.186907,9.6103 1.15,1.00,2.15,20,0.064550,0.061227,0.068040,0.061100,12.1851 1.15,1.00,2.15,30,0.014600,0.013028,0.016358,0.015103,14.1259 1.15,1.00,2.15,45,0.001650,0.001175,0.002316,0.001856,19.9188 1.15,1.00,2.15,60,0.000250,0.000107,0.000585,0.000228,19.1489 1.15,1.00,2.15,80,0.000000,0.000000,0.000192,0.000014,nan 2.15,2.00,4.15,2,0.865550,0.860752,0.870208,0.865333,2.2400 2.15,2.00,4.15,3,0.803400,0.797834,0.808849,0.804961,3.0374 2.15,2.00,4.15,5,0.696350,0.689940,0.702685,0.696559,4.1676 2.15,2.00,4.15,8,0.555700,0.548804,0.562575,0.560702,5.6364 2.15,2.00,4.15,12,0.419950,0.413126,0.426805,0.419854,7.0472 2.15,2.00,4.15,20,0.237300,0.231455,0.243246,0.235413,9.0937 2.15,2.00,4.15,30,0.116750,0.112373,0.121274,0.114221,11.1184 2.15,2.00,4.15,45,0.039250,0.036646,0.042031,0.038603,13.1697 2.15,2.00,4.15,60,0.012850,0.011380,0.014507,0.013046,14.8446 2.15,2.00,4.15,80,0.003300,0.002595,0.004196,0.003071,17.3209 3.15,3.00,6.15,2,0.907350,0.903253,0.911290,0.907029,1.7923 3.15,3.00,6.15,3,0.868700,0.863949,0.873310,0.863838,2.3150 3.15,3.00,6.15,5,0.781850,0.776073,0.787519,0.783526,3.3299 3.15,3.00,6.15,8,0.677500,0.670988,0.683944,0.676839,4.4464 3.15,3.00,6.15,12,0.557050,0.550155,0.563923,0.556837,5.7067 3.15,3.00,6.15,20,0.374150,0.367468,0.380880,0.376889,7.6897 3.15,3.00,6.15,30,0.223650,0.217928,0.229478,0.231377,9.1251 3.15,3.00,6.15,45,0.107600,0.103381,0.111970,0.111296,11.1475 3.15,3.00,6.15,60,0.054150,0.051098,0.057373,0.053535,12.3964 3.15,3.00,6.15,80,0.018250,0.016485,0.020200,0.020177,13.7362 #END BD #BEGIN BDTHRESH lambda,mu,turnover,q,N_ana,N_sim,SE,cap 0.65,0.50,1.15,0.769231,11.4182,11.5809,0.0821,160 1.15,1.00,2.15,0.869565,21.4345,21.5632,0.1543,160 2.15,2.00,4.15,0.930233,41.4229,41.6632,0.2544,192 3.15,3.00,6.15,0.952381,61.4003,60.6069,0.3655,284 #END BDTHRESH #BEGIN HORIZON key,G5,G10,G20,G50,G100,q pois,0.62095940,0.70256434,0.74134186,0.75088372,0.75099787,0.75099795 nb4,0.67498883,0.75203088,0.78795328,0.79693697,0.79705341,0.79705349 geom,0.77022878,0.83385306,0.86224627,0.86946047,0.86956512,0.86956522 nb025,0.89127393,0.92755097,0.94249344,0.94628699,0.94634798,0.94634805 lot30,0.98377952,0.98791632,0.98972340,0.99015241,0.99015744,0.99015744 #END HORIZON #BEGIN HALDANE key,variance,surv_exact,surv_haldane,N_exact,N_haldane pois,1.1500,0.249002,0.260870,10.4617,11.4836 nb4,1.4806,0.202947,0.202617,13.2067,14.7852 geom,2.4725,0.130435,0.121335,21.4345,24.6898 nb025,6.4400,0.053652,0.046584,54.3248,64.3084 lot30,33.1775,0.009843,0.009042,302.8648,331.3030 #END HALDANE #BEGIN SENS mean,Poisson,Geometric,Lottery30 1.02,65.5317,130.6033,1900.8010 1.05,30.2531,60.9446,876.8901 1.10,15.4613,31.4292,447.8631 1.15,10.4617,21.4345,302.8647 1.25,6.4534,13.4251,186.6069 1.50,3.4268,7.3884,98.7969 2.00,1.8798,4.3219,53.8645 #END SENS ================================================================================================ TOTAL RUNTIME: 174.5 seconds ================================================================================================